To Infinity and Beyond

Transcription

1 To Infinity and Beyond 22 January 2014 To Infinity and Beyond 22 January /34

2 In case you weren t here on Friday, Course website: Get a copy of the syllabus from that site. Contact Dr. Morandi if you have questions. The Canvas website for the course doesn t have course materials. An i>clicker2 is necessary for this course. Other models won t work. An app on a smartphone or a laptop can be used. To Infinity and Beyond 22 January /34

3 If you do not have a clicker, make sure you get the i>clicker2 and not an older model. right: wrong: To Infinity and Beyond 22 January /34

4 You can use a smartphone instead of an i>clicker2. If you wish to do this you need to download the app. There is a usage fee for using the app. See for more details. To Infinity and Beyond 22 January /34

5 Clicker Registration If you have not already done so, go to learn.nmsu.edu, logon, click on Math 210, and click on the Register your clicker link. To Infinity and Beyond 22 January /34

6 First Clicker Test Turn your clicker on and make sure the frequency is set to AA. If your s isn t, press and hold the power button until the set freq screen comes on. Then press A twice. You will receive the day s participation points by answering the following questions. There are no right or wrong answers; these are not quiz questions. To Infinity and Beyond 22 January /34

7 Please respond to the following question. Before this semester I used clickers A never B in one class C in more than one class To Infinity and Beyond 22 January /34

8 Multiple Choice We will do a sample multiple choice question. The clicker allows up to 5 responses. In which college are you enrolled? A Agriculture B Arts and Sciences C Business D Health and Social Services E None of the above You can change your answer before the test ends by hitting a new response. Only your last response gets saved. To Infinity and Beyond 22 January /34

9 Numeric Enter a number. It can be your age, your favorite number, a randomly chosen number, anything you want. You can enter decimals. You should learn how to do that. Once you hit your number, hit the send button. As with multiple choice, you can send multiple answers, but only the last one will be saved. To Infinity and Beyond 22 January /34

10 Alphanumeric What is your major? Please enter as an abbreviation as it appears in course listings. For example, ENGL or HIST or CJ. Again, after you finish typing, hit the send button. To Infinity and Beyond 22 January /34

11 If you have any questions about the clickers, ask me after class, in office hours, or me your question. We ll now start discussing issues about infinity. To Infinity and Beyond 22 January /34

12 Infinity The concept of infinity has both fascinated and frustrated people for millenia. We will discuss some historical problems about infinity, some modern (around 100 years old) ideas about infinity, and some interesting puzzles about the concept. To Infinity and Beyond 22 January /34

13 Zeno s Paradoxes There are a series of paradoxes possibly coming from the Greek philosopher Zeno ( BC). One we will discuss is the paradox of Achilles and the Tortoise. Aristotle is quoted referring to this paradox: In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. To Infinity and Beyond 22 January /34

14 Here is the statement of the paradox (borrowed in large part from Wikipedia): In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Suppose Achilles allows the tortoise a head start of 100 yards. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 yards, bringing him to the tortoise s starting point. During this time, the tortoise has run a much shorter distance, say, 10 yards. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. To Infinity and Beyond 22 January /34

15 Clicker Question Do you think Achilles will never catch up to the Tortoise? A Yes B No To Infinity and Beyond 22 January /34

16 A Football Paradox The 49ers and Cowboys are playing. The 49ers have the ball on the 1 yard line. As they start the play the Cowboys are called off sides. The penalty moves the ball half the distance to the goal line, so now it is on the 1/2 yard line. Again, the Cowboys are off sides; the penalty moves the ball half the distance to the goal line. They keep being called for off sides. No matter how many penalties, the ball is not quite to the goal line. If they get called for infinitely many penalties, shouldn t the 49ers end up in the end zone? To Infinity and Beyond 22 January /34

17 Galileo s Paradox To Infinity and Beyond 22 January /34

18 The following comes from Galileo s book Dialogue Concerning the Two New Sciences (from books.google.com) In readable form... To Infinity and Beyond 22 January /34

19 Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty. I take it for granted that you know which of the numbers are squares and which are not. To Infinity and Beyond 22 January /34

20 Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves. Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? Simplicio: Most certainly. Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. To Infinity and Beyond 22 January /34

21 Simplicio: Precisely so. Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together. To Infinity and Beyond 22 January /34

22 Sagredo: What then must one conclude under these circumstances? Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes equal, greater, and less, are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. To Infinity and Beyond 22 January /34

23 To put this more briefly, the paradox is that it seems that the set of all square numbers {1, 4, 9, 16, 25,...} is both smaller than and the same size as the set of all whole numbers {1, 2, 3, 4, 5,...}. Both cannot be true. Galileo gets the idea that these two sets may be the same size by seeing that elements can be paired off, without leaving out anything. To Infinity and Beyond 22 January /34

24 Giving a simpler example of this idea, Clearly if I have 10 M&Ms and you take 8 of them, then you have fewer M&Ms than I started with. Galileo s paradox makes this basic fact not clear for infinite sets. So, is the set of all square numbers smaller than the set of all whole numbers? Is it the same size? Does the question even make sense? To Infinity and Beyond 22 January /34

25 Georg Cantor To Infinity and Beyond 22 January /34

26 Cantor, a German mathematician, whose career was mostly in the later part of the 19th century, did important work in set theory. He formalized the idea of the size of a set, and defined what it means for one set to be larger, smaller, or the same size as another set. He used the idea Galileo discussed. His work, applied to infinite sets, was harshly criticized by many, including some of the most famous mathematicians of the time. One, David Hilbert, strongly supported his work. We will revisit Hilbert in the next class. To Infinity and Beyond 22 January /34

27 How to Compare Sizes of Sets Kids can compare two sets before knowing their numbers by pairing off elements. For example, a very young child can understand that there are just as many M&Ms as cars in the following picture. To Infinity and Beyond 22 January /34

28 Roughly, two sets have the same size if one can pair off elements of one set with elements of the other, leaving no elements left out of the pairing. One set is larger than another if the second is the same size as a subset of the first, but not vice-versa. To Infinity and Beyond 22 January /34

29 Infinite Sets Cantor extended this idea to arbitrary sets by saying two sets have the same size, whether or not they are finite or infinite, if elements of one can be paired with elements of the other, leaving no elements out. With his definition, it is true that the set of whole numbers {1, 2, 3, 4, 5,...} is the same size as the set of squares {1, 4, 9, 16, 25,...}, by what Salviati says in Galileo s dialogue. Cantor also defined a set to be infinite if it is the same size as a proper subset. It is not at all obvious that this corresponds to our intuition. For example, the set of whole numbers is infinite, because it is the same size as the set squares. To Infinity and Beyond 22 January /34

30 Galileo s paradox arises from the thought that one should be able to say two sets are the same size if you can pair up elements, which makes perfect sense for finite sets. Galileo thought that the set of squares shouldn t be as large as the set of whole numbers, even though they can be paired off. With Cantor s definition of infinity, any infinite set would be just as mysterious to Galileo. To Infinity and Beyond 22 January /34

31 Clicker Question Are all infinite sets the same size? A Yes B No If time permits we ll discuss this next time! To Infinity and Beyond 22 January /34

32 Back to Zeno Zeno s paradoxes can be resolved with the use of calculus, which makes sense of the notion of adding infinitely many numbers together. Zeno s paradoxes require one to think that adding infinitely many numbers together would result in an infinite sum. However, this is not always true. The football example may make this easier to see. No matter how many penalties the Cowboys incur, the total distance the ball gets moved is never more than 1 yard. If they got infinitely many penalties, then the distance would be exactly 1 yard. To Infinity and Beyond 22 January /34

33 Next Time We will look at some puzzles about infinity, coming from an idea of Hilbert, which is called the Hilbert Hotel. To Infinity and Beyond 22 January /34